Abstract

Monte Carlo simulation is used to determine the distribution of game lengths in number of moves for three popular children's games: Cootie, Candyland and Chutes and Ladders. The effect of modifications to the existing rules are investigated. Recommendations are made for preserving the sanity of parents who must participate in the games.

Introduction

I have the good fortune to be the father of a four-year-old. As such, I often play games designed for pre-schoolers. Some are better than others, and some seem to go on forever. While I enjoy the interaction with my daughter, sometimes I just want to get the game over with. The three most popular games in our house are Cootie, Candyland and Chutes and Ladders. The objective of each of these games is to be the first to finish. Progress towards finishing, however, is determined by chance (through the throw of a die, spinning a pointer or drawing a card) and there is a theoretical possibility that each of the games can last indefinitely. This study was undertaken to determine which games are the most and least likely to produce sessions of extraordinary length. Modifications of the existing rules, intended to shorten the games, are also explored.

Description of the Games

Cootie, Candyland and Chutes and Ladders have been around for a very long time. Many people of my generation played them when we were kids. For those of you in this group, the following paragraphs will refresh your memory, or if you haven't played the games, explain the basics. Most parents of preschool children can probably skip this section as they do not need to be reminded of the rules.

Cootie is the simplest of the games considered in this study. The objective of Cootie is to be the first person to build a complete cootie bug from parts supplied. As in all the games considered here, 2-4 players may participate. Players roll a single six sided die to determine which part they add to their bug. A complete bug consists of a body (roll 1), a head (roll 2), a tongue (roll 3), two eyes (roll 4), two antennae (roll 5) and six legs (roll 6). Players must start by acquiring a body and then a head before adding other parts. Players successfully adding a part get another roll of the die at their turn. If no part is added, the die is passed to the next player. Play continues until one player completes his cootie.

Chutes and Ladders is a board game where players spin a pointer to determine how they will advance. The board consists of 100 numbered squares. The objective is to land on square 100. The spin of the pointer determines how many squares the player will advance at his turn, with an equal probability of advancing from 1 to 6 squares. (Question: why not just use a die? Probably because a die is easier to lose. Also, the spinner livens up the game some when the pointer stops "on the line.") However, the board is littered with chutes, which move a player backward if landed on, and ladders, which advance a player. Chutes have pictures of bad behavior leading to disasters while ladders have pictures of good behavior leading to rewards. Most of the chutes and ladders produce relatively minor changes in position, however, several produce large gains (if you land on square 28 you advance to square 84) or losses (if you land on square 87 you go back to square 24).

Candyland is also a board game, consisting of 134 squares which alternate through yellow, red, orange, blue, purple and green. In addition, there are also two shortcuts (which advance the player), six special picture squares and 3 trap squares (see below). As in Chutes and Ladders, the goal is to finish the game by landing on the last square. Players advance by drawing cards which instruct them to move to the next square of a particular color, the second square of that color, or directly to one of the six special picture squares. Note that drawing one of the special picture cards can move a player forward or backward (though it seems as though it is usually backwards). If a player lands on one of the three trap squares, he or she must draw a card of the specified color before advancing.

Simulation of the Games

Monte Carlo simulations were developed of each of the games using MATLAB. Cootie, and Chutes and Ladder were particularly straightforward because the outcomes for each of the players is independent. Thus one can simulate games of one player, then randomly match pairs of games (or more if you are considering more players) and take the minimum number of turns as the length of the game. This can not be done for Candyland, however, because players draw cards (which determine the moves) from a common finite deck. Therefore Candyland must be simulated for a particular number of players. In each of the games moves are entirely determined by chance; there is no opportunity to make decisions regarding play. (This, of course, is one reason why most adults with any intellectual capacity have little interest in playing the games for extended times, especially since no money or alcohol is involved.)

Each of the games was simulated 200,000 times and the length of each game in number of turns was recorded. It was assumed that 2 players participated in each game. The normalized distributions of the game lengths are shown in Figure 1, panels A, B and C for Cootie, Candyland and Chutes and Ladders, respectively. The distributions are binned into segments of 5 moves. The mode, median and mean game lengths are shown in Table 1, along with the percentage of games of more than 75 and 100 turns. From the table it can be seen that Candyland games have the longest average length, while Chutes and Ladders games are typically the shortest. From the figures it is apparent that Cootie has the most narrow and normal distribution of game lengths. There is almost no chance that a game of Cootie will exceed 75 turns per player. In Chutes and Ladders, however, there is a 0.76% chance that the game will go over 75 moves.

The distribution even more skewed for Candyland, where there is a 3.4% chance a game will go over 75 moves, and a 0.77% chance the game will go over 100 moves (from personal experience I can say that a game of this length seems to last a very long time).

Based on this, it would appear that the best game (from a parent's perspective) would be Chutes and Ladders, closely followed by Cootie. However, using Monte Carlo simulation it is easy to investigate the effects of changing the rules of the games. In particular, I have investigated the effect of removing the longest chute (from square 87 to 24) from Chutes and Ladders and of changing the special picture square rule in Candyland. Specifically, the rule was modified so that a picture card only moved you forward, or not at all if the card indicated a special square which you had already passed. The game length distributions from the modified Chutes and Ladders and Candyland are shown in panels D and E of Figure 1, respectively (note that the vertical scale has been expanded in these last two panels relative to the first three). The mean, median and mode of the game lengths is included in Table 1. The change in Chutes and Ladders is certainly observable, but the change in Candyland is dramatic. In modified Chutes and Ladders there is only a 0.04% chance a game will exceed 75 moves. In modified Candyland, the chance of a game of over 75 moves has been effectively eliminated (no games over 75 moves were observed in 200,000 trials). Note however, that the most likely game length remains unchanged for Candyland and is actually one turn longer for Chutes and Ladders.

Table 1. Mean, Median and Mode of Game Lengths.

Mean

Median

Mode

% > 75

% >100

Cootie

27.7

27

26

0.02

0.0005

Chutes and Ladders

26.3

23

18

0.76

0.09

Candyland

33.3

29

18

3.4

0.77

Modified Chutes and Ladders

23.3

22

18

0.04

0.001

Modified Candyland

21.5

21

19

0.0

0.0

Conclusions

This article demonstrates how Monte Carlo simulation can be used to solve a real-world, every day problem: Of these three games, which one will provide entertainment for my four-year-old yet let me retain my sanity? If your child is inflexible regarding changing the rules, choose Cootie or Chutes and Ladders which have similar average game lengths. Of the two, Chutes and Ladders is probably the more interesting because of the possibility of moving both forward and backward. If your child insists on Candyland, consider changing the rules as suggested above. An alternative strategy, of course, is simply to let your child cheat. This not only shortens the games, but has the additional incentive that it usually causes the child to win and puts them in a better mood (though it certainly doesn't teach much about ethics). On the bright side, in the few weeks it has taken to complete this study, we have progressed from board games to card games, specifically Uno, which are much more interesting for adults and children. Perhaps there is a God after all.

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